Set of studied PLSs.
Results of theoretical, modeling, and experimental investigation of microwave acoustic properties of piezoelectric layered structure “Me1/AlN/Me2/(100) diamond” have been presented within a wide frequency band 0.5–10 GHz. The highest among known material quality parameter Q × f ~ 1014 Hz for the IIa type synthetic diamond at operational frequency ~10 GHz has been found. Conditions of UHF excitation and propagation of the bulk, surface, and Lamb plate acoustic waves have been established and studied experimentally. Frequency dependencies of the impedance and quality factor have been studied to obtain a number of piezoelectric layered structure parameters as electromechanical coupling coefficient, equivalent circuit parameters, etc. Results of 2D finite element modeling of a given piezoelectric layered structure have been compared with the experimental ones obtained for the real high-overtone bulk acoustic resonator. An origin of high-overtone bulk acoustic resonator’s spurious resonant peaks has been studied. Results on UHF acoustic attenuation of IIa-type synthetic single crystalline diamond have been presented and discussed in terms of Akhiezer and Landau–Rumer mechanisms of phonon–phonon interaction. Identification and classification of Lamb waves belonging to several branches as well as dispersive curves of phase velocities have been executed. Necessity of introducing a more correct Lamb-mode classification has been recognized.
- aluminum nitride
- synthetic diamond
- ultra-high frequency
- bulk acoustic wave
- Lamb wave
- electromechanical coupling
- acoustic attenuation
Known piezoelectric materials such as quartz, lithium niobate, lithium tantalate, langasite, etc. have been widely used as the substrates in a number of acoustoelectronic devices such as resonators, acoustic filters, delay lines, sensors, etc. Such devices can operate on one or several types of acoustic waves, such as conventional bulk (BAW) acoustic waves, surface (SAW) acoustic waves (Rayleigh, Sezawa, Gulyaev-Bleustein, shear-horizontal (
Well-known types of BAW resonators are the conventional piezoelectric resonators more often produced out of crystalline quartz, including the high-frequency resonator as inverse mesa structure, thin-film bulk acoustic resonators (FBAR) , and solidly mounted resonators (SMR) [1,2], but they have significantly lower both
The choice of HBAR’s substrate material is a problem of high importance since it is necessary to take into account an appropriate combination of physical and chemical properties, low acoustic attenuation at UHF, crystalline quality, possibility of precise treatment, etc. It is well known that physical properties of thin films such as the density and elastic constants can considerably differ from the ones measured on the bulk specimens. HBAR can be used as an instrument for determination of material properties of substrates and thin films at microwave frequencies with high accuracy. Additionally, HBAR application gives a unique possibility to measure frequency-sensitive properties, for example, the acoustic attenuation of substrate’s material within a wide frequency range of 0.5–10 GHz.
Application of diamond as a substrate material in this chapter is caused by its unique physical and acoustical properties: it is the hardest crystal with highest BAW and SAW velocities (in  direction, the phase velocity of BAW longitudinal type
The main objectives of this chapter consist of the theoretical, experimental, and modeling study of acoustic wave propagation, especially Lamb modes, and its dispersive properties in diamond-based piezoelectric layered structures.
2. Acoustic modes propagated in piezoelectric layered structures
2.1. Piezoelectric layered structure as a complex acoustic system: normal bulk, surface and plate acoustic waves
Propagation of the small amplitude acoustic waves in the piezoelectric crystal is described by the equations of motion and electrostatics and, additionally, the equations of state of the piezoelectric medium as :
There were introduced the values to be used as follows: ρ0 is the crystal density; is the unit vector of the dynamical elastic displacement;
Taking into account the dispersion of the elastic waves, a general form of the Christoffel equation and its components can be written as follows:
where is the wave vector and is the unit vector of the wave normal; α
The propagation of elastic waves in the multilayer piezoelectric structure should be written introducing additionally the boundary conditions depending on the number
one can obtain a matrix of the boundary conditions. Here, and are the amplitude coefficients of elastic displacements and electric potential in the
If we assume that the lower layer is thick enough (semi-infinite type), that is, its thickness is much greater than the length of the elastic wave, then in this case the latter equation in Eq. (3) can be ignored, that is, a free bottom border takes place. It is also necessary to require the fulfillment of the condition which provides attenuation of elastic waves in the substrate as . Then the equations describing the propagation of elastic waves in the "layer–substrate" structure will look like
Digital superscripts 1 and 2 denote the layer and the substrate, respectively; and
2.2. Dispersive relations for the elastic waves in plates and layered structures
Before investigation of the propagation of elastic waves in a multilayer structure such as “Me1/AlN/Me2/diamond,” it makes sense initially to study the marginal cases as propagation of elastic waves in a plate and a layered structure (interlayer interface).
Let us perform the analysis of the characteristics of an acoustic wave in piezoelectric crystalline plate belonging to the point symmetry 23. All the results obtained will be reasonable for other cubic crystals too. In this case, Christoffel tensor Eq. (2) written for the propagation of elastic wave along the  direction in (001) plane takes the following form:
Obviously, in this case Γ
Characteristic Eq. (7) with respect to
Roots of Eq. (8) take the following solutions:
where , ,
where and .
Equating to zero the determinant of the matrix (11), one can get the equation describing the propagation for the symmetric mode:
and for the antisymmetric mode of Lamb wave :
Here is modulus of the wave vector of the longitudinal bulk acoustic wave, and is the solution of biquadratic dispersive Eq. (8).
Solving Eq. (15) with respect to
Here is the modulus of wave vector of the shear bulk acoustic wave, and is the electromechanical coupling coefficient (EMCC). Eigenvectors corresponding to a value are equal to
As a result, in this case, the determinant of the boundary conditions (Eq. (5)) is obtained as follows:
Taking the similar procedure as for the Lamb wave solution, the matrix of the boundary conditions (18) is divided into two independent parts, and the dispersive equations describing the propagation of
It should be noted that in this case the
If we need to deposit a thin metal layer on the upper surface of the piezoelectric crystalline substrate without disturbing the mechanical boundary conditions, then the second row of the determinant (18) is changed, which leads to the following new dispersive equation for
In this case, an additional factor on the right side is the dynamic electromechanical coupling coefficient, depending on the frequency/plate thickness, and taking into account the effect of metallization on the magnitude of the phase velocity of piezoactive
Another marginal case is the propagation of an elastic wave when interlayer interface has been taken into account. Let us consider the propagation of the Love wave (
Here and after the superscript S is marked the substrate material constants. In an isotropic dielectric layer, due to the absence of the piezoelectric effect, the parameters of partial components are and , where . Here and after the superscript L is marked the layer material data. Consequently, the boundary conditions (6) can be written as
Equating to zero the determinant of Eq. (24), one can get the dispersive equation for the Love wave propagating in the layered structure “isotropic dielectric layer/piezoelectric crystalline substrate” :
where and .
Similar dispersive relation for Love waves (
2.3. Analysis of anisotropy and dispersion of SAW parameters in "AlN/diamond" PLS
Algorithm for calculating the elastic wave parameters in PLS was based on the method of partial waves, previously well adapted to define the SAW characteristics. To improve the accuracy of calculation, the normalization of Christoffel equations and boundary conditions was applied. Square of EMCC was determined by the relation
Using data on the elastic properties of diamond , and aluminum nitride (AlN) , the computer simulations of the SAW propagation in PLS “AlN/(111) diamond” were carried out with options of (100) or (001) on the AlN film orientation. Anisotropy of the Rayleigh wave phase velocity in the (111) plane of the diamond is relatively small, and the (001) plane of AlN film is isotropic with respect to the elastic properties; so, in PLS “(001) AlN/(111) diamond,” the SAW propagation is actually happening as on the isotropic medium. In contrast, the “(100) AlN/(111) diamond” PLS gives us an example of noticeable anisotropy in the phase velocity and EMCC.
Determination of what SAW type will be excited in this case is possible, considering the tensor of piezoelectric coefficients for the 6
Let us choose the SAW propagation X́2 along the  direction in the (100) plane of AlN film. AC electric field vector in accordance with the orientation of IDT will have the components located in the sagittal plane. Components of mechanical stress can be found by the relation , where , and . Last component is responsible for pure
Dispersion curves of SAW phase velocities and EMCCs for the “(001) AlN/(111) diamond” PLS depending on the
Dispersion curves of SAW phase velocities and EMCCs versus the
Figure 3 represents the anisotropy of SAW parameters in the “(001) AlN/(111) diamond” PLS. A set of curves is associated with data for the three values of the
Figure 4 shows the anisotropy of SAW parameters in the “(100) AlN/(111) diamond” PLS. As follows from Figure 4b, best EMCC values of 1.6 and 0.7% have a fundamental Rayleigh mode
3. Experimental Investigations of Acoustic Wave Excitation and Propagation in Diamond Based PLSs
3.1. Objects of Investigation
The IIa-type synthetic diamond single crystals grown by HPHT method at the Technological Institute for Superhard and Novel Carbon Materials were used as substrates for the studied PLSs. All the substrate specimens were double-side polished up to the roughness
|PLS #||PLS composition||Material/thickness
(nm) of top
of AlN film
(nm) of bottom
|Thickness (μm) of
The influence of area size and the shape of the electrodes on the signal quality was studied. Serial number of resonator’s notation is associated with the shape, while the letter represents the area of the top electrode. Here #“1” means a pentagon form, #“2”–irregular rectangle, and #“3”–circular form. The area of electrode “a” is 40000, “b”–22500, and “c”–10000 μm2. All these 1a–3c electrodes are associated with different HBARs with the same AlN and metal layers.
3.2. UHF study of acoustic wave excitation and propagation in piezoelectric layered structures
Microwave studies of PLSs were carried out by equipment (Figure 6) comprising the E5071C network analyzer (300 kHz–20 GHz) and the M-150 probe station. The experiments were fulfilled in the reflection mode with a test device connected by ACP40-A-SG-500 probe (the distance between tips was equal to 500 μm). Figure 7 shows the photograph of one of the studied samples (PLS #3).
3.3. PLS microwave acoustic properties measuring: phase velocities, quality factor and quality parameter, frequency and temperature dependences
PLS gives us a unique possibility to investigate the acoustic properties within a wide frequency band from several MHz up to tens of GHz: from fundamental λ/2 resonance (
The resonant frequencies of PLS can be calculated approximately by the relation:
Note that the resonance phenomena at low frequencies in PLS will be realized with low TFPT effectiveness.
It was found that such parameter as impedance
where should be measured in the frequency area away from the acoustic resonance. Only after this procedure, the proper
A full view of a number of measured PLS parameters such as
Frequency dependence of the extracted impedance |
Resonant peak at extreme operational frequency combining with high
Analyzing the data on the frequency dependence of
Analyzing Figure 14, an unusual frequency dependence of
Because the PLS in comparison with conventional piezoelectric resonator is an inhomogeneous acoustical device, a lot of unwished waves can be excited, distorting sometimes a useful signal. As an example here, we will focus only on 1a resonator (PLS #3). The amplitude–frequency characteristic (AFC) for this resonator is represented in Figure 15. As one can see, at rather low frequencies the main resonant peak has a number of adjacent so-called spurious peaks, which are placed at slightly higher frequencies. Such spurious peaks have lower amplitudes and are located at a certain distance away from the resonant signal. In this case, they have no influence on the main peak and its
Study of temperature dependence on resonant frequency for diamond-based PLS is also of great practical importance. The experimental results on the temperature dependence of normalized frequency
For a number of studied PLSs, the TCFs at different frequencies are represented in Table 2. The obtained TCF values were in a range(−4.5to − 6.5) × 10− 6
|PLS#||Resonant frequency, MHz||TCF, 10-6 K-1|
The typical temperature dependence of
In order to excite the SAW, a special interdigital transducer (IDT) should be formed. Study of SAW propagation was carried out by means of a delay line as the “Pt IDT/AlN/(001) diamond” PLS #8 (Figure 18). Here, one can see two identical SAW delay lines based on (100) diamond substrate 4 × 4 mm2, and including IDTs (period
3.4. UHF acoustic attenuation in diamond
The results on the frequency dependence of quality parameter can be explained in terms of acoustic attenuation. There are many different mechanisms of acoustical energy losses in PLS. Careful study shows that the attenuation in a rather thick diamond substrate is much higher than the one for thin electrode and AlN film . The influence of roughness losses of AlN film and diamond can be estimated as :
Fundamental origin of acoustical attenuation in solids is defined by phonon–phonon interaction, which can be concerned with two different mechanisms. At relatively low frequencies, when
As seen from Figure 20, the quality parameter has almost no frequency dependence while
4. Modeling of microwave acoustic properties of “Me1/AlN/Me2/diamond” piezoelectric layered structure
4.1. Equivalent circuit and frequency dependences of equivalent parameters for «Me1/AlN/Me2/diamond» PLS
HBAR’s equivalent parameters are close concerned with specified equivalent scheme of device and are important in view of HBAR modeling and application, for example, the AFC calculation, matching of devices, etc. PLS equivalent scheme introduced in Ref.  is represented in Figure 21.
Frequency dependence of equivalent parameters for the 1a, 1b, and 1c resonators (PLS #3) is represented in Figure 22. As one can see, there is a complex frequency dependence of equivalent parameters which requires an accurate choice of operational frequency bands.
4.2. Peculiarities of PLS acoustic wave excitation by thin film piezoelectric transducer
It is easy to show that as a rule the minimums of |
4.3. 2D FEM simulation results of acoustic wave propagation in diamond-based PLS
The 2D FEM simulations were carried out with the use of Comsol Multiphysics Simulation Software. Material data on the density, elastic constants, acoustic attenuation, etc. for all the layers and diamond substrate were taken from Refs. [16, 21, 30]. Such PLS parameters and acoustic processes as distribution of elastic displacement fields within all the parts of PLS (cross-sections of substrate and TFPT), AFR modeling, calculation of the wavelengths and phase velocities of acoustic modes of different types, identification of its types, etc. have been studied in detail. Symmetrical boundary conditions on the lateral borders of diamond substrate were used in order to satisfy the condition of zero normal displacement components as , where is a unit vector of the normal to the lateral border, and is a unit vector of the wave elastic displacement. Visualization of elastic displacement fields gave us an instrument of analysis of acoustic wave excitation because all the waves propagating in lateral directions should be reflected under the condition
In Figure 25, the patterns of elastic displacements for some acoustic waves are presented. One can clearly see the arrangement of displacement vectors designated by arrows. Color graphics serves, knowing the displacement along the Y-direction (up or down) and its magnitude. For example, if we take into account Rayleigh wave displacements (Figure 25c), there are ten λ
Effect of energy trapping was observed in the conventional piezoelectric resonators and was explained by authors  as a total internal reflection of acoustic beam on the vertical borders within the resonator’s aperture. Such effect was observed in HBARs too , but it has a more complex nature. As an example, the appearance of energy trapping in the PLS #3 is represented in Figure 26. It should be noted that realization of energy trapping is strongly concerned with the BAW wavelength, lateral dimensions, substrate thickness, and TFPT aperture. In the case of PLS #3, the energy trapping regime in the steady state was established, beginning the 12th BAW overtone at frequencies above ~268 MHz.
4.4. Identification, selection and classification of acoustic waves of different types
PLS has a sophisticated acoustic spectrum. Generally speaking there are three normal bulk acoustic waves propagating along any crystalline direction within a substrate: Rayleigh waves on free top and bottom substrate surfaces; symmetrical and antisymmetrical Lamb plate waves when the thickness of substrate is comparable to the wavelength. Presence of piezoelectric layer gives a possibility of excitation not only to bulk waves in vertical direction, but also to modified dispersive Rayleigh waves, including Sezawa wave with highest phase velocity, and Love waves, belonging to the
Type and order of the acoustic modes were defined, taking into account the directions of elastic displacement vectors, location, and the number of homogeneous areas of elastic displacements along vertical and horizontal axes when the fields of elastic displacements analogous to Figure 25 have been analyzed.
Dispersive Lamb waves are of special interest because there are a lot of types and orders of dispersive branches for its phase velocities. In order to distinguish Lamb modes, let us remember that the critical wavelengths (or critical frequencies) for each Lamb mode should be defined  as
4.5. Phase velocity dispersion curves and critical frequencies for acoustic waveguide Lamb modes
Dispersive properties of Lamb waves are important in a physical and practical sense. Based on 2D FEM simulation of PLS #3, all the possible modes have been observed within the 0–250 MHz band, than the mode’s identification and classification have been executed . As a result, the dispersion curves for the phase velocities for a lot of Lamb modes have been obtained (Figure 27). As one can see, the phase velocities of the
Phase velocities of all other
Analyzing a set of dispersive curves, one can notice that the difference between Lamb modes with respect to a mode creation’s point (e.g.,
Taking into account Eq. (38), one can see that critical frequencies for
In the similar manner the dispersive curves of phase velocities of Lamb waves can be obtained at other frequency bands and will be useful for preliminary analysis of design of advanced acoustoelectronics devices.
A lot of “Al/AlN/Mo/(100) diamond” PLSs have been studied both theoretically and experimentally within a wide frequency band of 0.5–10 GHz. At first time, the highest among known material quality parameter
This study was performed by a grant of Russian Science Foundation (project #16-12-10293).
Lakin K.M., Kline G.R., McCarron K.T. High-Q microwave acoustic resonators and filters. IEEE Transactions on Microwave Theory and Techniques. 1993; 41(12):2139–2146. DOI: 10.1109/22.260698
Mirea T., DeMiguel-Ramos M., Clement M., Olivares J., Iborra E., Yantchev V., et al. AlN solidly mounted resonators for high temperature applications. In: Proceedings of the Ultrasonics Symposium (IUS), 2014 IEEE International; 3–6 September; Chicago, IL. IEEE; 2014. p. 1524-1527. DOI: 10.1109/ULTSYM.2014.0377
Sorokin B.P., Kvashnin G.M., Volkov A.P., Bormashov V.S., Aksenenkov V.V., Kuznetsov M.S., et al. AlN/single crystalline diamond piezoelectric structure as a high overtone bulk acoustic resonator. Applied Physics Letters. 2013; 102(11):113507. DOI: 10.1063/1.4798333
Zhang H., Pang W., Yu H., Kim E.S. High-tone bulk acoustic resonators on sapphire, crystal quartz, fused silica, and silicon substrates. Journal of Applied Physics. 2006; 99(12):124911. DOI: 10.1063/1.2209029
Baumgartel L., Kim E.S. Experimental optimization of electrodes for high Q, high frequency HBAR. In: Proceedings of the Ultrasonics Symposium (IUS), 2009 IEEE International; 20–23 September; Rome. IEEE; 2009. p. 2107–2110. DOI: 10.1109/ULTSYM.2009.5441814
Mansfeld G.D., Alekseev S.G., Polzikova N.I. Unique properties of HBAR characteristics. In: Proceedings of the Ultrasonics Symposium, 2008 (IUS’2008). IEEE; 2–5 November; Beijing. IEEE; 2008. p. 439–442. DOI: 10.1109/ULTSYM.2008.0107
Sorokin B.P., Kvashnin G.M., Kuznetsov M.S., Burkov S.I., Telichko A.V. Influence of the temperature and uniaxial pressure on the elastic properties of synthetic diamond single crystal. In: Proceedings of the Ultrasonics Symposium (IUS), 2012 IEEE International; 7–10 October; Dresden. IEEE; 2012. p. 763–766. DOI: 10.1109/ULTSYM.2012.0190
Anthony T.R., Banholzer W.F., Fleischer J.F., Wei L., Kuo P.K., Thomas R.L., et al. Thermal diffusivity of isotopically enriched 12C diamond. Physical Review B. 1990; 42(2):1104–1111. DOI: 10.1103/PhysRevB.42.1104
Wells G.M., Palmer S., Cerrina F., Purdes A., Gnade B. Radiation stability of SiC and diamond membranes as potential x-ray lithography mask carriers. Journal of Vacuum Science & Technology B. 1990; 8(6):1575–1578. DOI: 10.1116/1.585118
Sorokin B.P., Telichko A.V., Kvashnin G.M., Bormashov V.S., Blank V.D. Study of microwave acoustic attenuation in a multifrequency bulk acoustic wave resonator based on a synthetic diamond single crystal. Acoustical Physics. 2015; 61(6):669–680. DOI: 10.1134/S1063771015050164
Aleksandrov K.S., Sorokin B.P., Burkov S.I. Effective piezoelectric crystals for acoustoelectronic, piezotechnics and sensors. Novosibirsk: SB RAS; 2007. pp. 501.
Burkov S.I., Zolotova O.P., Sorokin B.P., Aleksandrov K.S. Effect of external electrical field on characteristics of a Lamb wave in a piezoelectric plate. Acoustical Physics. 2010; 56(5):644–650. DOI: 10.1134/S1063771010050088
Burkov S.I., Zolotova O.P., Sorokin B.P. Calculation of the dispersive characteristics of acoustic waves in piezoelectric layered structures under the effect of DC electric field. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 2012; 59(10):2331–2337. DOI: 10.1109/TUFFC.2012.2458
Kessenich G.G., Lyubimov V.N., Shuvalov L.A. On surface Lamb waves in piezoelectric crystals. Crystallography Reports. 1982; 28(3):437–443.
McSkimin H.J., Andreatch P.Jr., Glynn P. The elastic stiffness moduli of diamond. Journal of Applied Physics. 1972; 43(3):985–987. DOI: 10.1063/1.1661318
Sotnikov A.V., Schmidt H., Weihnacht M., Smirnova E.P., Chemekova T.Yu., Makarov Y.N. Elastic and piezoelectric properties of AlN and LiAlO2 single crystals. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 2010; 57(4):808–811. DOI: 10.1109/TUFFC.2010.1485
Wu S., Ro R., Lin Z.-H., Lee M.-S. Rayleigh surface acoustic wave modes of interdigital transducer/(100) AlN/(111) diamond. Journal of Applied Physics. 2008; 104(6):064919. DOI: 10.1063/1.2986215
Wu S., Ro R., Lin Z.-X., Lee M.-S. High velocity shear horizontal surface acoustic wave modes of interdigital transducer/(100) AlN/(111) diamond. Applied Physics Letters. 2009; 94(9):092903. DOI: 10.1063/1.3093528
Sorokin B.P., Kvashnin G.M., Bormashov V.S., Volkov A.P., Telichko A.V., Gordeev G.I., et al. The manufacturing technology of the UHF transducers based on AlN film deposited on a synthetic diamond single crystal substrate. Izvestiya Vysshikh Uchebnykh Zavedeniy Seriya “Khimiya I Khimicheskaya Tekhnologiya”. 2014; 57(4):17–21.
Mansfeld G., Alekseev S., Kotelyansky I. Bulk acoustic wave attenuation in langatate. In: Proceedings of the 2001 IEEE International Frequency Control Symposium and PDA Exhibition; 06–08 June; Seattle, WA. IEEE; 2001. p. 268–271. DOI: 10.1109/FREQ.2001.956201
Sorokin B.P., Kvashnin G.M., Telichko A.V., Gordeev G.I., Burkov S.I., Blank V.D. Study of high-overtone bulk acoustic resonators based on the Me1/AlN/Me2/(100) diamond piezoelectric layered structure. Acoustical Physics. 2015; 61(4):422–433. DOI: 10.1134/S106377101503015X
Akhiezer A. On acoustic attenuation in solids. Journal of Experimental and Theoretical Physics. 1938; 8(12):1318–1329.
Landau L., Rumer Yu. Absorption of sound in solids. Physikalische Zeitschrift der Sowjetunion. 1937; 11(18):227–233.
Tabrizian R., Rais-Zadeh M., Ayazi F. Effect of phonon interactions on limiting the f.Q product of micromechanical resonators. In: Proceedings of the Solid-State Sensors, Actuators and Microsystems Conference, 2009. Transducers 2009. International; 21–25 June; Denver, CO. IEEE; 2009. p. 2131–2134. DOI: 10.1109/SENSOR.2009.5285627
Auld B.A. Acoustic fields and waves in solids, Vol. 1. New York, NY: Wiley; 1973. pp. 423.
Fitzgerald T.M., Silverman B.D. Temperature dependence of the ultrasonic attenuation in Al2O3. Physics Letters A. 1967; 25(3):245–247. DOI: 10.1016/0375-9601(67)90883-3
Mitsui T.V., Abe R., Furuhata Y., Gesi K., Ikeda T., Kawabe K., et al. Landolt-Börnstein, Zahlenwerte und Funktionen aus Naturwissenschaft und Technik. Neue Serie, Gruppe III: Kristall-und Festkörperphysik. Band 3: Ferro- und Antiferroelektrische Substanzen. Berlin-Heidelberg-New York: Springer-Verlag; 1969. pp. 584.
Wen C.P., Mayo R.F. Acoustic attenuation of a single-domain lithium niobate crystal at microwave frequencies. Applied Physics Letters. 1966; 9(4):135–136. DOI: 10.1063/1.1754679
Dieulesaint E., Royer D. Elastic waves in solids: applications to signal processing. New York, NY: Wiley; 1980. pp. 511.
Mansfeld G.D., Alekseev S.G., Kotelyansky I.M. Acoustic HBAR spectroscopy of metal (W, Ti, Mo, Al) thin films. In: Proceedings of the Ultrasonics Symposium, 2001 IEEE (Vol. 1); 07–10 October; Atlanta, GA. IEEE; 2001. p. 415–418. DOI: 10.1109/ULTSYM.2001.991652
Shockley W., Curran D.R, Koneval D.J. Trapped-energy modes in quartz filter crystals. The Journal of the Acoustical Society of America. 1967; 41(4B):981–993. DOI: 10.1121/1.1910453
Kvashnin G., Sorokin B., Telichko A. Resonant transformation of acoustic waves observed for the diamond based HBAR. In: Proceedings of the Frequency Control Symposium & the European Frequency and Time Forum (FCS), 2015 Joint Conference of the IEEE International; 12–16 April; Denver, CO. IEEE; 2015. p. 396–401. DOI: 10.1109/FCS.2015.7138866
Viktorov I.A. Rayleigh and Lamb waves: physical theory and applications. 1st ed. Springer, New York; 1967. pp. 154. DOI: 10.1007/978-1-4899-5681-1
Sorokin B.P., Kvashnin G.M., Telichko A.V., Novoselov A.S., Burkov S.I. Lamb waves dispersion curves for diamond based piezoelectric layered structure. Applied Physics Letters. 2016; 108: 113501.